System Analysis of HIV Infection Model with CD4+T under Non-Singular Kernel Derivative

Infectious diseases have caused the death of many people throughout the world for centuries. For this purpose, many researchers have investigated these diseases for establishing new treatment and protective measures. The most important of these is HIV disease. In this study, an HIV infection model of CD4+T cells is handled comprehensively with the newly defined Atangana-Baleanu (AB) fractional derivative. The existence and uniqueness of the solutions for fractionalized HIV disease model with the new derivative by considering the Arzela-Ascoli theorem.


Introduction
Over the past 50 years, the Human Immunodeficiency Virus (HIV) has become a lethal disease that affects the world in a global sense. HIV is a virus that generates infection by targeting immune system cells and causes fatal consequences if infection progresses. According to the researches conducted by the World Health Organization (WHO) worldwide, approximately 35 million people were affected by this disease, 940000 people died from HIV-related causes at the end of 2017. These statistics are inevitably increasing in spite of all the precaution taken over the years.
For this reason, many scientists worldwide are doing various researches in order to prevent such deadly diseases. Mathematical modeling is the most important part of these researches. With the help of mathematical models, researchers obtain very crucial information about the spread of diseases and measures to be taken [1][2][3].
In this paper, by the above motivation, we will handle an HIV infection model of CD4 + T cells, considered in [1], for investigating the system components under the effect of non-singular kernel derivative which is defined in [12].
For this purpose, the rest of the paper is divided into 4 Sections. In Section 2, some basic necessary definitions and theorem for Atangana-Baleanu (AB) derivative, which is also named as non-singular Mittag-Leffler kernel derivative, are given. Section 3 is devoted to the model construction. In Section 4, the existence and uniqueness of the solutions for the HIV infection model in the frame of the AB fractional derivative are given. Finally, in Section 5, our results are briefly summarized.

Basic definitions and preliminaries
Definition 1. Let f ∈ H 1 (a, b), a < b be a function and υ ∈ [0, 1]. The Atangana-Baleanu derivative in Caputo type of order υ of f is given by [12]: Definition 2. Let g ∈ H 1 (a, b), a < b be a function and υ ∈ [0, 1]. The Atangana-Baleanu derivative in Riemann-Liouville type of order υ of f is defined by [12]: Definition 3. The fractional integral related to the fractional derivative is given by [12]:

Model description
In this section, the fractional order HIV infection model of CD4 + T cells by the AB fractional derivative is constructed according to the reference in [1].
This system can be modeled by the three fractional order nonlinear differential equations as follow: where, T (t) is the number of healty CD4 + T cells at time t, I (t) is the concentration of infected CD4 + T cells by the HIV viruses at time t, V (t) is the number of HIV viruses at time t and υ, r, T max , k, β , µ and γ are the estimated data from real world applications.

Existence and uniqueness of the solutions
In this section, the existence and uniqueness of the solutions to the model (4) is described and proved under the effect of AB fractional derivative.
For this purpose, first of all, we apply AB fractional integral in Eq. (3) on both sides of the system and we get We can choose our kernels s (t, T (t)), s (t, I (t)), s (t,V (t)) for simplifying the demonstration of the equation (5) as follows, At first, we need to be able to identify an operator. We will then show that this operator is compact. Now, we consider the operator K : H → H and then we get Lemma 1. Let M ⊂ H be bounded in this way, we can find n, r, l > 0 such that Then K (M) is compact.
Proof. Let N = max 1−υ B(υ) + s (t, T (t)) and 0 ≤ T (t) ≤ P. For T (t) ∈ M, then we have the followings Let R = max 1−υ B(υ) + s (t, I (t)) and 0 ≤ I (t) ≤ Q. For In a similar way, From the Eqs. (7)-(9), the function K is bounded. Now, we will consider t 1 < t 2 and T (t) ∈ M. For ε > 0, if |t 2 − t 1 | < δ then we obtain and then Now, we will investigate the following: Putting Eqs. (11) and (12) in Eq. (10), we have and then we find With the same rule step by step, we can obtain the following for other two equations. For each ε > 0, we can find Proof. We should consider the fixed point of the operator K. We know that the operator K : H → H is completely continuous. Let T 1 ≤ T 2 , I 1 ≤ I 2 and V 1 ≤ V 2 then we get By a similar way, we obtain Hence K is increasing operator. By the help of conjecture, we have Km ≥ m and Kn ≤ n. So, the operator K : n, m → n, m is compact and continuous from Lemma 1. In that case, H is a normal cone.
To obtain uniqueness of the solutions, we begin the following steps.
which yields For the other components of the model, we find and Therefore, if the following conditions hold the mapping K is a contraction, which implies fixed point and thus the model has a unique positive solution.

Concluding remarks
In this work, we have examined the system response of the HIV infection model in [1] by modeling the AB fractional derivative in Caputo sense. The theoretical studies have shown that the solution of the discussed model in Eq. (4) is exist and unique under the AB fractional derivative.